My knowledge of trigonometry are still insufficient to resolve this problem. Any help would be greatly appreciated.
Solving for $\alpha$:
$$\tanα+2\tan2α+4\tan4α+8\tan8α+16\tanα=\cotα$$
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What are your thoughts? – rurouniwallace Jun 16 '13 at 17:43
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1It would be better if you show some of your work – Jun 16 '13 at 17:44
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2Is the equation right? $\tan\alpha+\dots+16\tan\alpha=$ – Américo Tavares Jun 16 '13 at 17:48
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2@RAsa, can you please the last part? Should not it be $8\tan8\alpha+16\tan16\alpha?$ – lab bhattacharjee Jun 16 '13 at 17:49
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3relevant: http://math.stackexchange.com/q/124528/73324 – vadim123 Jun 16 '13 at 17:54
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tanα+2tan2α+4tan4α+8tan8α+16tanα=cotα – Rasa Jun 16 '13 at 17:57
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OK, then please edit your question. – John Bentin Jun 16 '13 at 20:40
3 Answers
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As $\cos2A=\cos^2A-\sin^2A, \sin2A+2\sin A\cos A,$
$$\cot A-\tan A=\frac{\cos^2A-\sin^2A}{\cos A\sin A}=2\cot 2A$$
$$2\tan2\alpha+4\tan4\alpha+8\tan8\alpha+16\tan\alpha=\cot\alpha-\tan\alpha=2\cot2\alpha$$
$$\implies 4\tan4\alpha+8\tan8\alpha+16\tan\alpha=2(\cot2\alpha-\tan2\alpha)=2\cdot2\cot4\alpha$$
$$\implies 8\tan8\alpha+16\tan\alpha=4(\cdot2\cot4\alpha-\tan4\alpha)=4\cdot(2\cot8\alpha)$$
$$\implies 16\tan\alpha=8\cdot(\cot8\alpha-\tan8\alpha)=8\cdot2\cot16\alpha$$
$$\implies \cot16\alpha=\tan\alpha=\cot\left(\frac\pi2-\alpha\right)$$
$$\implies 16\alpha=n\pi+\frac\pi2-\alpha$$ where $n$ is any integer
$$\implies \alpha=\frac{(2n+1)\pi}{2\cdot17}$$
lab bhattacharjee
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@vadim123, please have a look into the edited version. Have you noticed how the multiple angle ratios of $\cot$ get cancelled? – lab bhattacharjee Jun 16 '13 at 18:09
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While there may be a more slick way to solve a problem like this, you can repeatedly apply $$\tan(2x)=\frac{2\tan(x)}{1-\tan^2(x)}$$ until you have a rational equation in $\tan(\alpha)$. I haven't checked how unwieldly the resulting rational equation gets, since there is some uncertainty if the equation has the intended coefficients.
2'5 9'2
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1@Rasa This is not the same as the equation in your title, when you look at the "degree 8" term. It also seems odd that you have left out a coefficient of 16 as in $16\tan(16\alpha)$, rather than $16\tan\alpha$, since the way you have it this term can be combined with the first to give $17\tan\alpha$. – 2'5 9'2 Jun 16 '13 at 18:03
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Iterating the identity $\cot(\alpha)-\tan(\alpha)=2\cot(2\alpha)$, we get $$ \cot(\alpha)-\tan(\alpha)-2\tan(2\alpha)-4\tan(4\alpha)-8\tan(8\alpha)=16\cot(16\alpha)\tag{1} $$ Therefore, using $(1)$, the equation in question becomes $$ \cot(16\alpha)=\tan(\alpha)\tag{2} $$ which, due to the periodicity of $\tan$, is equivalent to $$ n\pi+\frac\pi2-16\alpha=\alpha\tag{3} $$ That is, $$ \alpha=\frac\pi{34}(2n+1)\tag{4} $$
robjohn
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